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In physics and geometry, a catenary() is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, but it is not a parabola: it is a (scaled, rotated) graph of the hyperbolic cosine. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette,〔MathWorld〕 or, particularly in the material sciences, funicular.〔''e.g.'': 〕 Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691. Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, so that forces do not result in bending moments. In the offshore oil and gas industry, 'catenary' refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. ==History== The word ''catenary'' is derived from the Latin word ''catena,'' which means "chain". The English word ''catenary'' is usually attributed to Thomas Jefferson, who wrote in a letter to Thomas Paine on the construction of an arch for a bridge: It is often said 〔For example Lockwood, ''A Book of Curves'', p. 124.〕 that Galileo thought the curve of a hanging chain was parabolic. In his ''Two New Sciences'' (1638), Galileo says that a hanging cord is an approximate parabola, and he correctly observes that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45°. That the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result was published posthumously in 1669.〔Lockwood p. 124〕 The application of the catenary to the construction of arches is attributed to Robert Hooke, whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary.〔("Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society" by Lisa Jardine )〕 Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon. In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram〔cf. the anagram for Hooke's law, which appeared in the next paragraph.〕 in an appendix to his ''Description of Helioscopes,'' where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram〔The original anagram was "abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux": the letters of the Latin phrase, alphabetized.〕 in his lifetime, but in 1705 his executor provided it as ''Ut pendet continuum flexile, sic stabit contiguum rigidum inversum,'' meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch." In 1691 Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli.〔 David Gregory wrote a treatise on the catenary in 1697.〔 Euler proved in 1744 that the catenary is the curve which, when rotated about the ''x''-axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles.〔 Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.〔Routh Art. 455, footnote〕 ==Inverted catenary arch== Catenary arches are often used in the construction of kilns. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.〔 〕 The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation , which is a catenary if . While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a "weighted catenary" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.〔 and 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Catenary」の詳細全文を読む スポンサード リンク
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